I am not sure whether I understand your question, so let me know when the following is not what you asked for.
- E. Landau, Verallgemeinerung eines {\it Polya}schen Satzes auf algebraische Zahlkörper Gött. Nachr. 1918, 478--488
proved that the regulator R of a number field of degree n and discriminant D satisfies $R < c(n) \cdot \sqrt{D} \cdot \log^{n-1} D$. Landau's methods were improved upon by
- R. Remak, Elementare Abschätzungen von Fundamentaleinheiten und des Regulators eines algebraischen Zahlkörpers, J. Reine Angew. Math. 167 (1932), 360-378.